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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 7 Copyright © Orchard Publications Common Window Functions E.2.3 Hanning Window Function The Hanning * window function is defined as (E.6) and its time domain waveform is as shown in Figure E.12. Figure E.12. Hanning window function in time domain The Fourier transform of the Hanning window function for is (E.7) A typical amplitude response of the Hanning window function is shown in Figure E.13 and it was created with the MATLAB script below. fplot('20.*log10(abs(sin(pi.*x)./(pi.*x)).*(1./(1 (x+eps).^2)))',[0 5 50 0]) * This window function is also known as Hann window function or cosine squared window function. ft () Hann π t τ 2 cos 1 2 -- 1 2 π t τ -------- cos +   for |t| τ 2 -- < == 0 otherwise = 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 τ 1 = F ω 1 2 πω sin -------------------- 1 1 ω 2 -------------- =

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Appendix E Window Functions E 8 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Figure E.13. Typical amplitude response for the Hanning window function Figure E.14 shows the normalized frequency domain plot for the Hanning window function cre- ated with the hann MATLAB function, and the script below. n=50; w=hann(n); [W,f]=freqz(w/sum(w),1,512,2); plot(f,20*log10(abs(W)));grid Figure E.14. Normalized frequency domain plot for the Hanning window function created with MATLAB We can also use the MATLAB Window Visualization Tool. The script below generates the plot shown in Figure E.15. L=50; wvtool(hann(L)) 0 1 2 3 4 5 -50 -40 -30 -20 -10 0 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 9 Copyright ©

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